## Nov 20, 2009

### An Ant and a Cube

An ant starts eating a 3*3 rubik's cube made up of cheese at a corner(vertex). What is the probability that the last cube it eats is the body-center cube?

The ant can only travel from a cube to the adjacent cubes (i.e. having common faces)

Courtesy: Nitin Basant

1. You cannot use any graph theory here, and neither can you use an expression that comes into balance when iterated. Suppose the cubes are as follows:

1 2 3
4 5 6
7 8 9

10 11 12
13 14 15
16 17 18

19 20 21
22 23 24
25 26 27

Suppose the ant's path is 1 2 3 6 5 4 7 8 11 12: Cube 9 will now drop down, which will change the connectivity of the graph. Therefore, the only way I can see to solve it is to write a recursive function that will generate all possible routes through the entire 3x3x3 cube. If, as a guess, there are an average of 2.5 choices from each position, then there will be several billion possible routes - so solving this puzzle would represent some serious work. Good luck to anyone who tries it! Moby Dick.

2. There's a very beautiful solution to the problem. Although the lengthier approach would also lead to the same solution.

This problem may seem very tough in the beginning but truly speaking it isn't. Here's a hint for you all- It's not possible that the last cube could be body-center cube.

But now explain how is it true?

3. If the cube beneath the body centre cube is eaten, then the body centre cube will drop down, and hence will no longer be the body centre cube. Moby Dick.

4. Paint the 27 little cubes alternately black or white (like a chess board, only three dimensional).
Suppose the corner little cube, with the ant, is black. Note that then there are 14 black and 13 white little cubes and (most importantly) the central cube is white. The ant eats alternatively black/white/black/white etc. Once it has eaten 13 black and the corresponding 13 white, there will be one BLACK remaining. So it is NOT the central one, which is white.
Thus, the central cube is never the last one.

ML from Greece

5. @Freddo: the rubik's cube doesn't have any cube at its body-center.